A note on an extension of Shidov's theorem and Shlyk's theorem

Abstract

Let $G$ be a non-solvable group and $p$ any fixed prime divisor of $|G|$. We prove that the intersection of all non-nilpotent maximal subgroups of $G$ of order divisible by $p$ containing the normalizer of some Sylow subgroup is $p$-closed, which provides a further generalization of Shidov's theorem and Shlyk's theorem.